with \(\rho _\gamma \)  the present Cosmic Microwave Background energy density. Within the standard \(\Lambda \) CDM cosmological model, this parameter is fixed to \(N_{\rm eff}=3.044\)    [1]}, [2]}, [3]}, [4]}, [5]}, [6]} which consists of three massless neutrino species and an additional correction coming from the non-instantaneous neutrino decoupling. Here we regard \(N_{\rm eff}\)  as an additional free parameter of the \(\Lambda \) CDM \(+\)  \(\Omega _K\)  \(+\)  \(N_{\rm eff}\)  cosmological model. It should be noted that when testing departures from its reference value, one can probe and constrain several extended models both of cosmology and particle physics that predict extra dark radiation in the early Universe, including the cases of additional neutrino species and hot relics beyond the standard model of elementary particles [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}.
